Stochastic programming problems depending on random elements only through the corresponding probability measure are considered in the paper. If this probability measure is substituted by its statistical estimate then, of course, only a statistical estimate of the optimal value and optimal solution can be obtained. The aim of the paper is to study the asymptotic behaviour of such estimates. The presented approach is based on the extended delta method. First, the definition of the problem as well as the definition of Hadamard differentiability and the basic delta method are recalled in the paper. Further, problems with deterministic constraint set are studied. The Lagrangian function is employed for the problems in which constraints can depend on the probability measure, too. The asymptotic analysis of the optimal value is presented very carefully while asymptotic properties of the corresponding optimal solution are discussed only briefly. Most of the presented results are proved separately in an appendix.